Any group of index 2 is normal [duplicate]
Solution 1:
If $ H $ is a subgroup of index 2, then it has two cosets in $ G $, which we may denote by $ H $ and $ G - H $. If $ g \in H $, then clearly we have $ gH = Hg = H $. Otherwise, $ gH, Hg \neq H $ since $ g \notin H $, however any coset of $ H $ is either $ H $ or $ G - H $, so we must have that $ gH = Hg = G - H $. Since the left and right cosets coincide, we conclude that $ H $ is normal.